arXiv subjects

Haohua Tang

Publications and source records attributed to Haohua Tang.

4 recordsLinked to original sources

arXiv2026

Online Matrix Factorization, Online Private Query Release, and Online Discrepancy Minimization

In this paper we consider several related online computation problems. First, we study answering sequences of statistical queries arriving online, and being answered immediately when they arrive with differential privacy. Known matrix factorization mechanisms can answer a set of statistical queries with error bounded by the $\gamma_2$ norm of their query matrix, but require that all queries are known in advance. We show that nearly the same error bounds can be achieved in the online setting for non-adaptively chosen queries. To do so, we give an online factorization algorithm that competitively matches the best offline factorization up to logarithmic factors. In the online matrix factorization problem, a new row $q_t$ of a matrix arrives at each time step $t$, and the algorithm needs to maintain a factorization $L_tR_t=Q_t$ such that at each time it appends some rows to $R_t$, and outputs a new row $\ell_t$ s.t. $\ell_tR_t=q_t$. Our algorithm maintains the competitiveness over this online process, even if the number of rows to arrive is unknown. As another application, we give an online discrepancy minimization algorithm that achieves discrepancy competitive against the $\gamma_2$ norm (and also against hereditary discrepancy) up to logarithmic factors.

cs.DS↗

arXiv2025

Weighted Fourier Factorizations: Optimal Gaussian Noise for Differentially Private Marginal and Product Queries

We revisit the task of releasing marginal queries under differential privacy with additive (correlated) Gaussian noise. We first give a construction for answering arbitrary workloads of weighted marginal queries, over arbitrary domains. Our technique is based on releasing queries in the Fourier basis with independent noise with carefully calibrated variances, and reconstructing the marginal query answers using the inverse Fourier transform. We show that our algorithm, which is a factorization mechanism, is exactly optimal among all factorization mechanisms, both for minimizing the sum of weighted noise variances, and for minimizing the maximum noise variance. Unlike algorithms based on optimizing over all factorization mechanisms via semidefinite programming, our mechanism runs in time polynomial in the dataset and the output size. This construction recovers results of Xiao et al. [Neurips 2023] with a simpler algorithm and optimality proof, and a better running time. We then extend our approach to a generalization of marginals which we refer to as product queries. We show that our algorithm is still exactly optimal for this more general class of queries. Finally, we show how to embed extended marginal queries, which allow using a threshold predicate on numerical attributes, into product queries. We show that our mechanism is almost optimal among all factorization mechanisms for extended marginals, in the sense that it achieves the optimal (maximum or average) noise variance up to lower order terms.

cs.DS↗

arXiv2023

General Gaussian Noise Mechanisms and Their Optimality for Unbiased Mean Estimation

We investigate unbiased high-dimensional mean estimators in differential privacy. We consider differentially private mechanisms whose expected output equals the mean of the input dataset, for every dataset drawn from a fixed bounded $d$-dimensional domain $K$. A classical approach to private mean estimation is to compute the true mean and add unbiased, but possibly correlated, Gaussian noise to it. In the first part of this paper, we study the optimal error achievable by a Gaussian noise mechanism for a given domain $K$ when the error is measured in the $\ell_p$ norm for some $p \ge 2$. We give algorithms that compute the optimal covariance for the Gaussian noise for a given $K$ under suitable assumptions, and prove a number of nice geometric properties of the optimal error. These results generalize the theory of factorization mechanisms from domains $K$ that are symmetric and finite (or, equivalently, symmetric polytopes) to arbitrary bounded domains. In the second part of the paper we show that Gaussian noise mechanisms achieve nearly optimal error among all private unbiased mean estimation mechanisms in a very strong sense. In particular, for every input dataset, an unbiased mean estimator satisfying concentrated differential privacy introduces approximately at least as much error as the best Gaussian noise mechanism. We extend this result to local differential privacy, and to approximate differential privacy, but for the latter the error lower bound holds either for a dataset or for a neighboring dataset, and this relaxation is necessary.

math.ST↗

arXiv2021

Near Neighbor Search via Efficient Average Distortion Embeddings

A recent series of papers by Andoni, Naor, Nikolov, Razenshteyn, and Waingarten (STOC 2018, FOCS 2018) has given approximate near neighbour search (NNS) data structures for a wide class of distance metrics, including all norms. In particular, these data structures achieve approximation on the order of $p$ for $\ell_p^d$ norms with space complexity nearly linear in the dataset size $n$ and polynomial in the dimension $d$, and query time sub-linear in $n$ and polynomial in $d$. The main shortcoming is the exponential in $d$ pre-processing time required for their construction. In this paper, we describe a more direct framework for constructing NNS data structures for general norms. More specifically, we show via an algorithmic reduction that an efficient NNS data structure for a given metric is implied by an efficient average distortion embedding of it into $\ell_1$ or into Euclidean space. In particular, the resulting data structures require only polynomial pre-processing time, as long as the embedding can be computed in polynomial time. As a concrete instantiation of this framework, we give an NNS data structure for $\ell_p$ with efficient pre-processing that matches the approximation factor, space and query complexity of the aforementioned data structure of Andoni et al. On the way, we resolve a question of Naor (Analysis and Geometry in Metric Spaces, 2014) and provide an explicit, efficiently computable embedding of $\ell_p$, for $p \ge 2$, into $\ell_2$ with (quadratic) average distortion on the order of $p$. We expect our approach to pave the way for constructing efficient NNS data structures for all norms.

cs.DS↗